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Cyclic inequality with real numbers close to each other

Source: VJIMC 2018, Category I, Problem 3

April 14, 2018
inequalitiesalgebra

Problem Statement

Let nn be a positive integer and let x1,…,xnx_1,\dotsc,x_n be positive real numbers satisfying ∣xi−xj∣≤1\vert x_i-x_j\vert \le 1 for all pairs (i,j)(i,j) with 1≤i<j≤n1 \le i<j \le n. Prove that x1x2+x2x3+⋯+xn−1xn+xnx1≥x2+1x1+1+x3+1x2+1+⋯+xn+1xn−1+1+x1+1xn+1.\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1} \ge \frac{x_2+1}{x_1+1}+\frac{x_3+1}{x_2+1}+\dots+\frac{x_n+1}{x_{n-1}+1}+\frac{x_1+1}{x_n+1}.
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